A180389 -id:A180389 - cp's OEIS frontend

A180388 Number of permutations of 1..n with number of rises (p(i+1)>p(i)) different from number of rises in the inverse permutation.

0, 0, 0, 0, 2, 24, 228, 2088, 19732, 197890, 2131060, 24729108, 309027560, 4148115048, 59611031304, 913915160628, 14897443573244, 257369478424890, 4698494631208500, 90393611677083892, 1828153614233855024, 38778554791649355232, 860921156414834534368

Offset: 0

Leroy Quet, D. S. McNeil and R. H. Hardin in the Sequence Fans Mailing List Sep 01 2010

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

a(n) + A180389(n) = n! = A000142(n).

More terms from Vaclav Kotesovec, Jun 10 2015

a(0)=0 prepended by Alois P. Heinz, Jun 10 2015

A298248 Triangle of double-Eulerian numbers DE(n,k) (n >= 0, 0 <= k <= max(0, 2*(n-1))) read by rows.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 4, 0, 1, 1, 0, 10, 2, 10, 0, 1, 1, 0, 20, 12, 54, 12, 20, 0, 1, 1, 0, 35, 42, 212, 140, 212, 42, 35, 0, 1, 1, 0, 56, 112, 675, 880, 1592, 880, 675, 112, 56, 0, 1, 1, 0, 84, 252, 1845, 3962, 9246, 9540, 9246, 3962, 1845, 252, 84, 0, 1

Offset: 0

Christian Stump, Jan 16 2018

DE(n,k) = number of permutations with d descents and e descents of the inverse such that d+e = k.

			The triangle DE(n, k) begins:
n\k 0    1     2     3      4      5      6     7     8    9   10
0:  1
1:  1
2:  1    0     1
3:  1    0     4     0      1
4:  1    0    10     2     10      0      1
5:  1    0    20    12     54     12     20     0     1
6:  1    0    35    42    212    140    212    42    35    0    1

Christian Stump, On bijections between 231-avoiding permutations and Dyck paths, MathSciNet:2734176

Dominique Foata and Guo-Niu Han, The q-series in Combinatorics; permutation statistics
FindStat - Combinatorial Statistic Finder, The sum of the number of descents and the number of recoils of a permutation

Row sums give A000142.

Cf. A000292, A008292, A180389.

q = var("q")
[sum( q^(pi.number_of_descents()+pi.inverse().number_of_descents()) for pi in Permutations(n) ).coefficients(sparse=False) for n in [1 .. 6]]

cp's OEIS Frontend

A180388 Number of permutations of 1..n with number of rises (p(i+1)>p(i)) different from number of rises in the inverse permutation.

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A298248 Triangle of double-Eulerian numbers DE(n,k) (n >= 0, 0 <= k <= max(0, 2*(n-1))) read by rows.

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